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arXiv · 1911.04925

Congruences on Square-Classes for the Partition Function

Martin Raum
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Abstract

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes. We thus elucidate for the first time an exemplary family of congruences found by Atkin-O'Brien. Our results are based on a novel framework that leverages available results on integral models of modular curves via representations of finite quotients of $\mathrm{SL}_2(\mathbb{Z})$ or $\mathrm{Mp}_1(\mathbb{Z})$. This framework applies to congruences of all weakly holomorphic modular forms.

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Martin Raum. 2019-11-12. Congruences on Square-Classes for the Partition Function. https://arxiv.org/abs/1911.04925

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